Advances in Differential Equations

Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$

Thierry Cazenave, Flávio Dickstein, and Fred B. Weissler

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Abstract

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the Navier-Stokes system with no external force in $\mathbb R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, for initial values $u_0$ small in a certain sense, we show that if the sequence of dilations $\lambda _n u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $\sqrt{t} u(t,\cdot\sqrt t)$ converges uniformly on $\mathbb R^N $ to ${{\boldsymbol{\mathcal S}}}(1)z$ along the subsequence $t_n=\lambda _n^2$, where ${{\boldsymbol{\mathcal S}}}(t)$ is the Navier-Stokes flow. If $N=2$ or $3$, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of an infinite-dimensional space. The resulting ``universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. Moreover, for a fixed $t_0>0$, ${{\boldsymbol{\mathcal S}}}(t_0)$ followed by an appropriate dilation generates a chaotic discrete dynamical system.

Article information

Source
Adv. Differential Equations, Volume 10, Number 4 (2005), 361-398.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867870

Mathematical Reviews number (MathSciNet)
MR2122695

Zentralblatt MATH identifier
1208.35101

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B40: Asymptotic behavior of solutions 37D45: Strange attractors, chaotic dynamics 37L99: None of the above, but in this section 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

Citation

Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B. Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$. Adv. Differential Equations 10 (2005), no. 4, 361--398. https://projecteuclid.org/euclid.ade/1355867870


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